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First Semester [B.Tech] – December 2003
Paper Code : ETMA – 101 Subject : Applied Mathematics  I 
Time : 3 Hours 
Maximum Marks : 75 
Note : Attempt any 5 questions in all. At least two questions from each section. All questions carry equal marks. 
SECTIONA
Q1 
( a ) 
State and prove Lagrange's Mean value theorem and verify it for the function f(x) = sin x + cos x in [0,π/2]. 
5 
( b ) 
If y = (sin^{1} x)^{2}, prove that (1 –
x^{2}) y_{2 } xy_{1} = 2 and also show
that 
5 

( c ) 
If ρ_{1}
and ρ_{2} are the
radii of curvature at the extremities of two conjugate diameters
of the ellipse 
5 
Q2 
( a ) 
Use Taylor's theorem to evaluate √10 correct to four significant figures. 
5 
( b ) 
Evaluate:

5 

( c ) 
Determine the asymptotes of 4x^{3}  3xy^{2}  y^{3} + 2x^{2}  xy  y^{2}  1 = 0 
5 
Q3 
( a ) 
Show that

5 

( b ) 
Test for the convergence of the following series : 

5 

( c ) 
The velocity v(km/min) of a moped, which starts from rest is given at fixed intervals of time. Estimate approximately the distance covered in 20 minutes by applying Simpson's 1/3 rule. 
5 


t 
2 
4 
6 
8 
10 
12 
14 
16 
18 
20 



v 
10 
18 
25 
29 
32 
20 
11 
5 
2 
0 

Q4 
( a ) 
Determine the length of the loop of the curve 9y^{2} = (x3)(x6)^{2} 
5 
( b ) 
Find the area common to the cardiode r = a(1+cos θ) and the circle r = (3/2)a and also the area of the remainder of the cardiode. 
5 

( c ) 
If the curve r = a+b cos θ (a>b) revolves about the initial line, show that the volume generated is (4/3) π a (a^{2} +b^{2} ) 
5 
SECTIONB
Q5 
( a ) 
If v = (x^{2} +y^{2} +z^{2} )^{1/2} show that

5 
( b ) 
If
(ii)

5 

( c ) 
Expand sin(xy) in powers of (x1) and (yπ/2) up to and including second degree terms. 
5 
Q6 
( a ) 
Find the volume of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1 
5 
( b ) 
Sketch the region of integration and evaluate:

5 

( c ) 
Changing into polar coordinates and hence evaluate:

5 
Q7 
( a ) 
Solve:

5 
( b ) 
Solve : ( 2xy cos x^{2} – 2xy +1 )dx + ( sin^{2 } x^{2 })dy = 0 
5 

( c ) 
Solve :

5 
Q8 
( a ) 
If u_{1}=(x_{2} x_{3})/x_{1} , u_{2}=(x_{1} x_{3})/x_{2} , u_{3}=(x_{1} x_{2})/x_{3} , then evaluate J ( u_{1}, u_{2}, u_{3}). 
5 
( b ) 
Solve :

5 

( c ) 
Solve :

5 